29. Infinite Values of a Logarithm. If, in the construction of Art. 23, A and become larger than any previously assigned arbitrarily large value, while their ratio m (that is, the modulus) remains unchanged, P and Q are transported instantly to an indefinitely great distance, and OP, OQ become simultaneously larger than any assignable magnitude.' It is customary to express this fact in brief by writing

though to suppose these values actually attained would require both A and μ to become actually infinite. This supposition will be justifiable whenever we find it legitimate, under the given conditions, to assign to the indeterminate form μ / λ ∞ ∞ a determinate value m.* In like manner, since from 6° =1 / ∞ = o, the equations

b

81

b- o, logmo=

=

∞

we may infer

are employed as conventional renderings of the fact, that when P and Q are moving to the left, P passing from J towards O and Q negatively away from O, x, in the equations xby, y=logm x, remains positive and approaches o, while y is negative and approaches

∞.

30. Indeterminate Exponential Forms. When v logm u becomes either o×∞, or ±∞ Xo, it is indeterminate (Art. 11). Now logm u is o if u 1, is +∞ if u = +∞, and is ∞ if u = o (Art. 29). Hence vX logm u will assume an indeterminate form under the following conditions:

* This form of statement must be regarded as conventional. Strictly speaking we cannot assign a value to an indeterminate form. When the quotient x/y, in approaching the indeterminate form, remains equal to, or tends to assume, a definitive value, we substitute this value for the quotient and call it a limit. In conventional language the indeterminate form is then said to be evaluated.

But if v × logm u, or logm uʊ is indeterminate, so is uʊ, and therefore the forms I ∞, o° are indeterminate.

∞

Whenever one of these forms presents itself, we write yu and, operating with ln, examine the form

If then In y can be determined, y can be found through the equation

VIII.

y=eln y.

SYNOPSIS OF LAWS OF ALGEBRAIC OPERATION. 31. Law of Signs:

*

(i). The concurrence of like signs gives the direct sign, + or X. Thus :

to

=+, XX=X, //=X.

(ii). The concurrence of unlike signs gives the inverse

+

or /. Thus:

-+= X1 = 1, 1X = 1.

(iii). The concurrence of two or more positive, or an even number of negative factors, in a product or quotient, gives a positive result. Thus :

(+ a) × (+b) = + a× b, (a) (b) = + axb.

(iv). The concurrence of an odd number of negative factors, in a product or quotient, gives a negative result. Thus:

(+ a) × (—b) = — a × b, (a) × (+b) = — axb.

* Cf. Chrystal: Algebra, vol. I, pp. 20-22.

=+ (±a) × (±c) + (±a) × (±d) +(±6) × (±c) + (± b) × (±d).

(ii). For division:

(±a±b)/(±c)=+(±a)/ (±c)+(±b)/(±c). A divisor consisting of two or more terms cannot be distributed over the dividend.

35. Laws of Exponents:

xn w

(i). Involution: (a) == (a×n)×m ̧

(ii). Index law: ama" = a±m±n ̧

Also, as a consequence of these two:

(iii). Corollary: (ab) = xam xbm.

36. Laws of Logarithmic Operation:

Zero may be regarded as the origin of additions, unity as the origin of multiplications.

38. Agenda: Involution and Logarithmic Operation in Arithmetic.

(1). Show that, if n be an integer, the index law (Art. 35) leads to the result:

(2). Show, by the law of involution and the index law (Art. 35), that 81/3- 2, 81/4

3, etc.

(3). Show, by the law of metathesis (Art. 36), that 2log 32 = 5, 3log 729=6, etc.

(4). Show, by the law of metathesis and the addition theorem (Art. 36), that

(5).

Find the logarithms: of 16 to base 22, of 125 to base 5 X 52, of 128 to modulus 1/ln 8, of 1/81 to modulus I / In 27.

CHAPTER II.

GONIOMETRIC AND HYPERBOLIC RATIOS.

IX. GONIOMETRIC RATIOS.

39. Definition of Arc-Ratio. In the accompanying figures N'N is a straight line fixed in position and direction, OP is supposed to have reached its position by turning about the fixed point O in the positive sense of rotation from the initial position ON. Any point

N'

CLAN

N'

A N

Fig. 21.

on OP at a constant distance from O describes an arc AVQ, a linear magnitude. Let the ratio of this arc to the radius OQ, both taken positively, be denoted by @, that is,

=(length of arc AVQ)! (line-segment OQ).

The amount of turning of OQ, that is, the angle AOQ, fixes the value of this ratio; and since the arcs of concentric circles intercepted by common radii are proportional to those radii (Prop. 18), the ratio may be replaced by an arc CD provided only OC be taken equal to the linear unit.

In